Sequential Methods for Coupled Geomechanics and Multiphase Flow

90 CHAPTER 4. CONVERGENCE OF THE DRAINED AND UNDRAINED SPLITS
(Strang, 1988), where Λ = diag {γe,1, γe,2} and P is an invertible matrix. When a fixed
iteration number, k = niter, is used, the error estimate of the drained split is
where e n+1,0
fs
= x n+1
f
e n+1,niter
fs
≤ (max|γe|) niter
e n+1,0
fs
− xn,niter s . Thus, (max|γe|) is equivalent to D.
Remark 4.3. From Equation 4.7, e n,niter
fs
, (4.26)
does not disappear even though ∆t approaches
zero, because (max|γe|) niter does not approach zero (i.e., O(1)). Thus, the drained split
with a fixed number of iterations is not convergent. Non-convergence of the drained split
becomes severe when max|γe| approaches one, which is also the same as the stability limit.
Remark 4.4. D is less than one if max|γe| ≤ 1 for all θ, which yields the stability
condition of the drained split during iterations. In order to have max|γe| ≤ 1 for all θ in
Equation 4.24, the stability condition is τ = b 2 M/Kdr ≤ 1, where τ is the coupling strength.
This stability condition has been obtained by the Von Neumann method in Chapter 3.
4.2.2 Comparison with the coupled flow and dynamics
The governing equations of the coupled flow and dynamics are given, for example, in Armero
and Simo (1992) as
··
Div σ + ρbg = ρbu,
(4.27)
˙m + Div w = ρf,0f, (4.28)
where ··
() is the second-order time derivative. Denote ˙u by vk, which is the rate of the solid
skeleton displacement. Then, Equations 4.27 and 4.28 can be expressed as
˙ξ = Aξ + s, (4.29)